Light
Light is something that has long fascinated mankind. As we will see later on, measuring its properties is something that ancient civilizations first had a hand in studying, before we were able to understand more about its nature. And- as we will see at the end of this section- we still do not fully understand the true nature of light. But, that comes later.
In earlier (as-yet-unfinished) sections of these notes, it will have been discussed that light is an electromagnetic wave, driven by waves of electric and magnetic fields. It is also from this combination of fields that it can be shown that the speed with which light moves is 𝑐 = 3 ⋅ 108𝑚/𝑠, or about 1.86 ⋅ 105 𝑚𝑖/𝑠, or fast enough to circle the globe about seven and a half times in one second. (You read this correctly. Light does move this swiftly.)
(Note: Do not worry if you do not know all of the background in the paragraph above. We can understand much about light and how it behaves without knowing much about its origins.) With that out of the way- trusting that light has properties similar to a wave- we might ask what forms light can take. Recalling from our section on waves the relationship between wave speed, frequency of the wave, and the wavelength of the wave- 𝑣 = 𝜆𝑓- we can ask ourselves what kinds of wavelengths or frequencies light can take. Below is a picture of the entire breadth of the “electromagnetic spectrum,” which allows us to categorize light into a few different “bands,” or categories:
Image credit: Sapling Learning: https://sites.google.com/site/chempendix/em-spectrum
stations that play all the Top 40 biggest hits, blare vitriol across talk shows, or report news. All radio stations register that they will broadcast at a frequency of, say, “93.9 𝑀𝐻𝑧 on the FM dial,” and no one else in the area can (legally) broadcast on that frequency. A radio host’s vocal chords emit sound waves, which stimulate a microphone, which disturbs an electric current, which is converted into radio waves, which are broadcast across the air. Those waves disturb your radio antennae, causing electric signals to flow through your speaker wire, which cause the speaker to vibrate, which then allows you to hear the host’s voice.
Moving up to the next band, we have microwaves. If you have already made the connection, yes- these are the same waves that are emitted from your microwave. (Or, to give the appliance its proper name, microwave oven.) Your microwave emits microwaves, which vibrate water molecules in the food, which cause them to heat up, which cooks your food.
Subsection – Light, Reflections, Radios, and Microwave Ovens You will notice that, if your microwave is properly shielded, the door to your microwave should have a mesh with holes no more than a millimeter or so across. Have you ever wondered why this is? Think back to the wave section, where waves interfere with boundaries. In order for a wave to pass through a gap in space- such as the holes in your screen, or a tunnel in a mountain- the wave must have a wavelength that is smaller than the gap in the screen or the width of the tunnel. If the wavelength is larger than the gap, then the wave will reflect off the boundary.
Take how we have classified microwaves: they have wavelengths on the order of 10−3 − 10−1 𝑚, or about 1 𝑚𝑚 to about 10 𝑐𝑚. If the gap in the shielding on your microwave is larger than about a millimeter in diameter, the shielding may not be doing its job properly. Check the wavelength (or frequency, then calculate the wavelength) of your magnetron, and compare that wavelength to the aperture of your shield. If the shield mesh has larger holes than the magnetron’s wavelength, it is time to replace your microwave!
The same effect is seen when you drive into a tunnel, and your radio signal goes dead. Did you ever notice how this happens more for the AM radio stations, and less so for the FM radio stations? For some basic information, AM stands for “amplitude
This difference means that AM waves tend to be on the order of hundreds of meters, while FM waves tend to be on the order of 1-10 meters in length. As a result, AM waves almost always die away when you enter a tunnel, because a tunnel tends not to be hundreds of meters wide, while FM radio waves tend to be able to slot through most tunnels.
As has always been the case in our course, there is a reason we do things the way we do, and those reasons are fueled by physics!
Light, cont’d At any rate, we return to our discussion of the spectrum.
Above microwaves, we have infrared light, which you may more commonly know as “IR” or “heat vision.” The formal term for light of these wavelengths is “invisible radiant energy.” It is invisible (just as radio, micro, X, UV, and 𝛾 rays are invisible) to our eyes because our eyes are not sensitive to these wavelengths, as we will see shortly. With that said, we often use IR cameras to “see” infrared light, which can let us determine where there is an energy leak in a system. (There are other uses, of course, just this is a very common example.) Though not all thermal energy need be emitted in this range, in the case of the Sun’s emissions, over half of all light the Sun emits is in the infrared spectrum.
Above infrared light, we finally have visible light- the small portion of the spectrum we can see. (You will notice this sliver is so small as to need to be “blown up” to see the whole “rainbow.”) The colors go from red, to orange, to yellow, to green, to blue, to violet, forming our familiar spectrum: ROYGBV. We will revisit this part of the spectrum in more detail later on.
Above violet, we have ultraviolet (UV) light. We are likely familiar with this kind of light because we usually wear sunscreen or sunblock to protect ourselves from these kinds of light rays. The reason for this is simple: as we have gone to higher and higher in frequency, the energy of the corresponding light rays has also increased. As is discussed in a later light section, rays of high enough energy are able to cause electrons to leave atoms (“ionizing radiation”), which can cause changes in the way those atoms behave. This has been shown to have a causative link with cancer. This is why from here on out, we try to limit our exposure to such rays as much as we can.
Above X-rays, we finally arrive at the highest-energy rays we have yet discovered: gamma (𝛾) rays. Their wavelengths overlap somewhat with the X-ray portion of the spectrum, though in physics, we define them by their origin. While X-rays are emitted by electrons bound to atoms, gamma rays are emitted by the nuclei of atoms. We tend to see gamma rays emitted in extreme situations, such as the splitting of an atom (atomic fission) or the joining of atoms (atomic fusion). We see each in the stars, supernovae, and black hole formation, in addition to engineering feats such as nuclear power plants and weapons.
During most of this section, we will deal with just the visible portion of the spectrum, though we will touch on some of the more intriguing aspects of light (which involve the other portions of the spectrum) in later sections of this text. We will also be dealing with three different models of light at various points throughout the text:
1) Ray model: this model allows us to represent a piece of light as a “ray.” Light will move in straight lines, with constant speed, in specific directions.
2) Wave model: this model allows us to represent multiple pieces of light as a “wave front,” letting us see the global behavior of light. Each wave front will move in a straight line, with constant speed, in specific directions.
3) Particle model: this model allows us to represent a piece of light as a “particle.” Each particle can move in straight lines with constant speed, but will be allowed to interact with other nearby pieces of light (or, even, itelf).
Most of this chapter’s emphasis will be on the ray and wave model, though in the last sections, we will discuss the particle model in some detail.
But, now that we are familiar with the breadth of the spectrum and how we can use models to represent light, we must ask ourselves a critical, basic question:
Why do we see?
How an Image Forms
The short explanation for why we see: “light enters our eyes and we see.” But it’s a little more complicated than that.
It would be better to say that when light is emitted from an object, some of that light enters our eyes, and we perceive that object along a line drawn from our eye straight to that object.
It should stand to reason that if the room is dark, the person will not see anything. So, let us turn on the flashlight:
It stands to reason that light emitted from the lightbulb in the flashlight will spread out in a cone, defined by the shape of the head of the flashlight. The person in this room will only see that light as being emitted from the flashlight if light reaches the person’s eyes. We can represent the light leaving the flashlight by any number of light “rays” leaving the flashlight, like this:
We could draw an infinite number of light rays leaving the flashlight, since the light is spreading out across that entire cone. However, we would like to be able to solve problems with this method of drawing rays, so we are going to choose our rays carefully. Only rays emitted from the flashlight that reach the person’s eye are important, here, since this is how the person will see the flashlight in the dark room. This means that there are a finite number of rays that hit this person’s eyes directly, which we can represent by connecting one position with the other:
So, the person is able to see the flashlight sitting precisely where it is in our diagram.
This idea of simplifying an infinite number of rays down to a small number of carefully-chosen rays will make our problems so much easier to solve in this section. Besides: do you really want to spend all your time drawing the infinite (and I do mean infinite) number of rays that could be drawn in every diagram?
As I thought.
The Law of Reflection and Fermat’s Principle
Suppose that there were an observer in a dark room, on the opposite side of a barrier. Let us send a single light ray from some flashlight at some angle toward a mirror, such that the person will see the flashlight. We will represent any mirror in this text as a line representing the reflective side of the mirror, while the “backside” of the mirror is represented by hashes. We can see the situation described in this diagram:
At what angle should the light from this flashlight hit the mirror, in order to reach the person? This diagram may seem obvious:
𝜃 𝜃
But is it so obvious? And is it actually what happens? (It is… but why?)
(Note: If you would like to read up on this matter in more detail, I highly recommend you read “QED” by Richard Feynman for a much more detailed treatment on the matter.)
First, we will note that the dashed line I have drawn is a normal line, with respect to the surface of the mirror. This means, as with the force chapter, that the line is perpendicular to the surface of the mirror. We will also define the angle that the light ray makes with the mirror as it comes into the mirror (or, is “incident upon” the mirror) as the “incidence angle,” 𝜃𝑖, and we will define the angle that the reflected light ray makes with the mirror as it leaves the mirror as the
“reflection angle,” or 𝜃𝑟. From the above diagram, it would appear that 𝜃𝑖 = 𝜃𝑟. This is true, and it is called the Law of Reflection. But the question of why it is true is very interesting.
Very clearly, the angles are dissimilar. If you were to measure the path length for this diagram versus the equal-angle situation, however, you would find that in this situation, the path length for the equal-angle situation is shorter, and is therefore traversed in a smaller amount of time. (Go ahead, break out the ruler and measure- you can confirm this for yourself! I will wait here.)
Let us take it to an even further extreme- consider this diagram:
Certainly, in this situation, the angles of incidence and reflection are nowhere near one another. But, also certainly, the path length that the light travels is much, much longer than when they are equal to one another. As a result, light takes the path that corresponds to angles of incidence and reflection that are equal to one another.
This is called Fermat’s Principle1
, or the principle of least time, which states that light will take whatever path that corresponds to the least time taken to travel between two points.2
To prove it another way, consider that diagram, again: (Note: it is not necessary to read this proof to understand the Law of Reflection and apply it correctly. Students who are familiar with calculus will have an easier time understanding this proof than others.)
𝑥 𝑑 − 𝑥
𝜃𝑖 𝜃𝑟
𝑎 𝑏
𝑑
The flashlight and person are separated by a distance 𝑑, each a distance of 𝑎 and 𝑏 from the mirror respectively, and light from the flashlight strikes the mirror a horizontal distance of 𝑥
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Brief history: Euclid and Ptolemy both came up with some basic understanding of reflection circa 300-60 BCE, and Ibn al-Haytham described an early version of this principle in 1021 CE. Pierre de Fermat- a famous
mathematician- expressed this relationship in a letter circa 1662 CE. It was found later that this principle cannot stand on its own, and itself relies on the Huygens-Fresnel Principle, which itself stated that every point which a light wave/ray/particle reaches will behave as a spherical source of light. Using this principle, the Law of Reflection (a.k.a. Fermat’s Principle) can be proven to be true.
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along the mirror, which means that the horizontal distance left for the light to travel is 𝑑 − 𝑥. In order to find the total path length for the light to travel, we need to find the hypotenuse of each triangle created by the light ray:
𝐿 = 𝑥2+ 𝑎2 + 𝑑 − 𝑥 2+ 𝑏2
In order to find path length that makes 𝐿 as small as possible, we apply what is called the first-derivative test, and take a first-derivative of this expression with respect to 𝑥- the arbitrary distance that the light travels before striking the mirror. (If you do not understand how calculus,
derivatives, chain rule, or the “first-derivative test” work, either read the notes about calculus, or skip down three lines of algebra.) We must use Chain Rule here, and so define 𝑢 = 𝑥2+ 𝑎2
and 𝑣 = 𝑑 − 𝑥 2+ 𝑏2, we find:
𝑑𝐿 𝑑𝑥 =
𝑑 𝑑𝑥 𝑢
𝑑 𝑑𝑢 𝑢 +
𝑑 𝑑𝑥 𝑣
𝑑 𝑑𝑣 𝑣
𝑑𝐿
𝑑𝑥 = 2𝑥 1 2
1
𝑥2+ 𝑎2 + 2 𝑑 − 𝑥 −1
1 2
1
𝑑 − 𝑥 2+ 𝑏2
𝑑𝐿 𝑑𝑥=
𝑥 𝑥2+ 𝑎2+
𝑥 − 𝑑 𝑑 − 𝑥 2+ 𝑏2
Now, applying the first-derivative test means setting all of this equal to zero, so:
𝑥 𝑥2 + 𝑎2+
𝑥 − 𝑑
𝑑 − 𝑥 2+ 𝑏2 = 0
Which means:
𝑥 𝑥2+ 𝑎2 =
𝑑 − 𝑥
𝑑 − 𝑥 2+ 𝑏2
If we look carefully, we could express the lefthand side as simply the sine of the incidence angle, because this is a ratio of two sides of a triangle; we can do the same for the righthand side, which would give us the sine of the reflection angle. Which means:
sin 𝜃𝑖 = sin 𝜃𝑟
Which means:
𝜽𝒊= 𝜽𝒓
Now, with a basic understanding of how sight, light, and reflection all work, we turn to another basic question: how do you see yourself in the morning, when you look into a mirror?
Plane Mirrors and an Introduction to Ray Diagrams
There are a few kinds of mirrors that we will deal with in this course. The first- and, arguably, the simplest- takes the form of a flat piece of glass with a reflective material placed on one side, otherwise known as a plane mirror. How do we see ourselves in such a device? Let us draw a ray diagram, or a diagram that shows a select number of light rays that allows us to determine how an image forms. For a person standing in front of a mirror, this diagram could look like:
How is this person able to see their feet in the mirror? Light must have been emitted from their feet, bounced off of the mirror, and into their eyes. Let us draw this ray. When we do so, we must follow the law of reflection, which- in this case- means that the ray reflects off a point that is equidistant between the person’s feet and eyes:
In order to conclude where an image forms, however, we will need at least one other ray emitted from this person’s feet to be reflected off the mirror. This ray does not necessarily need to end up in the person’s eye for us to determine where the image of the person forms. (In fact, following the Law of Reflection, light hitting the mirror at any other location cannot reach the person’s eye.) We could choose whatever angle we want, and in fact, we could choose any one of an infinite number of angles, and we would be able to find the image. It would, however, be easier to draw if we knew precisely what angle to use. For convenience, then (and only for
convenience, not because it has any special meaning), we will draw a light ray leaving the foot, impacting the mirror, and reflecting directly back toward the person: and incidence angle of 0°,
and a reflection angle of 0°.
inclination, lines drawn along those paths could meet to the right, or behind, the mirror. Let us extend these lines in our diagram. (At this step, I have removed the floor, such that you can see the reflected light rays along the floor a little better.)
The point where these rays intersect is where the person will see their feet! So, we add the image of their feet to the diagram:
If we repeat this whole process for the waist, we can determine the location of the waist! We will erase the rays for the feet, but leave everything else where it is, and begin again, following the same process:
And, following the same process, we would have the full image:
Here, I have labeled 𝑑𝑜 as the “object distance,” or the distance from the mirror to the object, and
wonder if this is always the case, and consider what happens if the mirror is curved in order to find out.
Concave Mirrors and the Mirror Equation
Suppose we were to take the mirror from the previous section and curl it in, like so:
This is a “concave” mirror, curved in this case to resemble a section of a circle or sphere. We could imagine that there is a point some distance away from this mirror that represent the center of such a circle or sphere, which we will denote as 𝐶, or the “center of curvature,” positioned a distance equal to the radius of curvature away from the mirror:
𝐶
We could imagine what happens when light is shone into this mirror by drawing it one ray at a time:
Note: it is important to remember that this only works well in situations where the angles of incidence and reflection in such mirrors (and, later, lenses) are small, or less than ~15°. When the angles involved are bigger, an effect called “spherical aberration” occurs, which results in an image that is blurry and colored oddly at the edges. For this whole section, we will assume that the angles are small. To read more, you can consult HyperPhysics3.
These diagrams is rather crudely drawn in Microsoft Word, but we might be able to convince ourselves from this diagram that all of the reflected light rays do not meet at 𝐶, but at 𝐶/2 away from the mirror. Because the light meets at this location- or “focuses” at this point- we call this the focal point of the mirror, 𝑓.
Now, how does such a mirror form an image? We could stick a person at some distance away from the mirror, draw some rays, and find out. This time, however, we will not draw a whole stick figure. Instead, we will represent “any” object you like by using a stand-in arrow, like so:
𝐶 𝑓
Let us draw a line connecting the mirror, the focal point, and the radius of curvature. This line is important- it is called the principal axis, or the optical axis, and is a line which passes through a mirror (or lens, as we will see later) about which the mirror (lens) is symmetric.
𝐶 𝑓
In order to figure out where the image of this object forms in this mirror, we must repeat the process of drawing rays and reflecting them off of the mirror. But now, what rays should we choose? There are an infinite number of rays that we could draw that are being emitted from this object, all at slightly different angles. It is tedious to have to draw an infinite number of anything. So, let us select a few carefully-chosen rays that will be much easier to draw. We have already shown that if we send a light ray straight into the mirror, parallel to the axis, that light ray will reflect through the focal point:
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It should stand to reason, then, that any light ray that passes through the focal point will reflect off the mirror parallel to the axis. This is simply the reverse of the previous ray. Our diagram gets modified to include this ray:
We could stop here, but we will add a third ray as a “check.” If we look at the center of the mirror, and zoom in enough, it would resemble a plane mirror that is standing straight up. This would mean that if we send a light ray into the mirror, it will reflect off with an angle that is fairly easy to measure with a protractor. (Note: you need only ever draw two rays to determine the location of an image. The third is usually used as a check.) So, we do this:
Where all of the reflected rays converge is where the image forms. This means that our diagram, and then our image, looks like this:
𝑑𝑜 𝐶 𝑑𝑖 𝑓
Suppose we took the diagram above and made triangles of this shape:
The triangles each of bases of 𝑑𝑜 and 𝑑𝑖, while their heights have lengths of ℎ𝑜 and ℎ𝑖,
mirror are the same, which means that all of their angles are the same. Therefore, they must be similar! This means that we can take ratios of sides:
ℎ𝑜 𝑑𝑜
= ℎ𝑖 𝑑𝑖
Stated another way,
ℎ𝑖 ℎ𝑜 =
𝑑𝑖 𝑑𝑜 = 𝑀
Here, 𝑀 stands for the magnification of the image. In a strict sense, the distance ratio should have an extra negative sign, standing for the fact that the image height in this case should be negative, but if we are only concerned about the relative size of the image compared to the object, this will do.
Now, suppose we were to draw two more triangles in that diagram:
One triangle has a height that is equal to the height of the object, which we will call ℎ𝑜, and the other has a height of the image, which we will call ℎ𝑖. They have bases of 𝑑𝑜− 𝑓 and 𝑓, respectively. Because of their geometry, they are similar triangles, which means that ratios of sides of one triangle should have equal ratios of corresponding sides for the other triangle. This would mean:
𝑀 = ℎ𝑖 ℎ𝑜 =
𝑑𝑖 𝑑𝑜 =
𝑓 𝑑𝑜 − 𝑓
If we were to consider just terms involving 𝑑𝑜, 𝑑𝑖 and 𝑓, we could multiple across, distribute, and combine terms, we would find:
𝑑𝑖𝑑𝑜− 𝑑𝑖𝑓 = 𝑑𝑜𝑓
𝑑𝑖𝑑𝑜 = 𝑑𝑜 + 𝑑𝑖 𝑓
Let us divide both sides by 𝑑𝑖𝑑𝑜:
1 = 𝑑𝑜+ 𝑑𝑖 𝑓 𝑑𝑖𝑑𝑜
1 𝑓 =
1 𝑑𝑜+
1 𝑑𝑖
This is called the Mirror Equation, used for mathematically solving all of our optics problems. We can reapply this formula to the example we used earlier: the object distance, according to Word, is about 6.680 𝑐𝑚, and the focal length is about 2.286 𝑐𝑚. Let us solve for the image distance:
1 𝑓=
1 𝑑𝑜 +
1 𝑑𝑖
1 𝑓−
1 𝑑𝑜 =
1 𝑑𝑖
𝑑𝑜 − 𝑓 𝑑𝑜𝑓
= 1
𝑑𝑖
𝑑𝑖 = 𝑑𝑜𝑓 𝑑𝑜− 𝑓 =
6.680 𝑐𝑚 2.286 𝑐𝑚
6.680 𝑐𝑚 − 2.286 𝑐𝑚 = 𝟑. 𝟒𝟕𝟓 𝒄𝒎
When I measure this in Word, I find that 𝒅𝒊= 𝟑. 𝟒𝟖𝟎 𝒄𝒎. This would indicate that my drawing has an error on the order of 0.1%. Perhaps Word’s drawing software is not too shabby, after all.
Let us reconsider what this image is. Contrast the image formed by this concave mirror with that formed by a plane mirror. In a concave mirror- at least, in this configuration- the light rays actually converge at some point, while in a plane mirror, the light rays never truly converged- only their paths did. This allows us to draw a distinction between types of images: virtual and real images.
A real image is an image that is formed by the convergence of real, actual light rays. You can project such an image onto a screen, because the photons (rays) actually meet somewhere.
A virtual image is an image that is formed by the convergence of light paths, but not real, actual light rays. You cannot project such an image onto a screen, because the photons (rays) do not actually meet. You must instead look into the optical device to see the image.
Now, where does the image form? We can draw the same rays, following the same process, as before:
Notice here, in this step, that the light ray that passes through the focal point need not have originated from beyond the focal point- as long as its path can form a straight-line path to the focal point, the optical pattern works. So, let us find the image!
Notice that this image is upright, while the previous image was inverted. This tends to be, but is not always, a clue that we have made a virtual image in this situation. We can also see that the image has become magnified in size. This is why concave mirrors tend to be used as shaving and makeup mirrors: they make smaller details easier to see, which is important to prevent cuts when shaving or errors when applying makeup. Let us run through the mathematics: the object
Let us begin by drawing the rays we have grown accustomed to:
Note that we cannot pass a ray from the object through the focal point, as the object is at the focal point. So, we try another:
If we look carefully, we will see that these reflected rays are parallel. So, where does the image form? Or does one form? After all, in math class, you were probably taught that parallel lines never meet… right?
… Or do they?
Think about it this way. In this series of drawings, I show pairs of lines that are progressively inclined less and less with respect to one another. In your mind’s eye, draw where they meet with one another. Then, consider the diagram above one more time.
They converge farther and farther away. So, at the limit when they are actually parallel to one another, they converge at infinity. Parallel lines do converge, they just do so at a distance of infinity.
Let us show this another way, by using the mirror equation:
1 𝑓=
1 𝑑𝑜
+ 1
𝑑𝑖
If the object distance equals the focal point, then:
1 𝑓 =
1 𝑓+
0 = 1 𝑑𝑖
The only value of 𝑑𝑖 that makes this equation true is if 𝑑𝑖 is infinite. (This is the meaning of your calculator saying “undefined” when you divide by zero- it is undefined because the result is infinity.)
So, is this a real, or virtual image? If the rays actually converge, this must mean that the image is real, inverted, and infinitely large in height. We can use this arrangement to project clear images onto a screen, no matter how far away that screen is. (Think drive-in movie theaters, for
example.) Or, we can project a column of light toward infinity… like a lighthouse.
To recap this section:
- You can calculate the magnification of an image using 𝑀 = ℎ𝑖 ℎ𝑜 =
𝑑𝑖 𝑑𝑜
- You can calculate the focal length of a mirror, the distance from the mirror to an object, or the distance from a mirror to an image by using 1
𝑓 = 1 𝑑𝑜+
1 𝑑𝑖
- Concave mirrors can form real or virtual images, depending on the situation.
- If you need to draw a ray diagram, try to draw any two of these three rays, first: 1) Draw a ray parallel to the axis; reflect the ray through the focal point of the mirror. 2) Draw a ray that passes through the focal point; reflect the ray parallel to the axis. 3) Draw a ray to the center of the mirror; reflect the ray at 𝜃𝑖 = 𝜃𝑟.
Convex Mirrors
Convex mirrors, in contrast to concave, bend outward instead of inward, like this:
Those reflected rays are diverging from a specific point, and we already know its name: the focal point of the mirror. We can trace these light rays back behind the mirror to find the focal point, and using the focal point, find the radius of curvature of the mirror:
Which, when I measure it in Word, the focal length is about 1.27 𝑐𝑚, and the radius of curvature is about 2.54 𝑐𝑚. Knowing this information, we can stick an object in front of the mirror (say, at a distance of 3.63 𝑐𝑚) and analyze the situation to find how an image forms in such a mirror. We use the exact same ray-tracing process described in the previous section, beginning with a ray drawn parallel to the axis, traveling through the focal point:
Notice that for this first ray, we have extended the ray to the focal point, to make sure our ray points in the right direction. The next ray ought to pass through the focal point, but the light ray will encounter the mirror before that happens. This is okay. The light ray, by geometry, will still pass parallel to the axis after being reflected, like so:
We could draw the third ray into the center of the mirror, and in general, it is good practice to do so. However, for our purposes, we will simply extend the already-drawn reflected rays to find the location of the image:
run the math, we should find that 𝑑𝑖 = −1.95 𝑐𝑚. Again, the negative denotes a distance behind the mirror. This yields about 40% error, again revealing the limitations of Word’s drawing software. If more care had been taken in constructing the diagram, or if the focal length involved had been made a bit larger, there would have been a smaller margin of error. Try it yourself, with your own pencil, ruler, protractor, and compass!
Subsection: Focal Length of a Plane Mirror
Let us revisit our first ray diagram, involving a plane mirror. We determined that 𝑑𝑜 = −𝑑𝑖. So, what is the focal length of such a mirror? Let us substitute this into our mirror equation, and find out!
1 𝑑𝑜
+ 1
𝑑𝑖
= 1 𝑓
1 −𝑑𝑖
+ 1
𝑑𝑖
=1 𝑓
0 = 1 𝑓
This can only be true… if the focal length is infinite. This must be the case! Imagine shining parallel light beams straight into a flat mirror. These light rays, when reflected off the mirror, will meet… an infinite distance away from the mirror.
How remarkable, that the mirror equation applies to any type of mirror!
Refraction, the Greeks, and Snell’s Law (a.k.a. Fermat’s Principle, Redux) Consider your normal, everyday (transparent) beverage, like water. If you put a straw into it, how does it look, to your eye? In order to answer this question, we must allow light to pass from one medium to another- through objects, rather than just reflect off of single objects. The
Greeks- specifically, Ptolemy- asked questions about how this worked, and even did experiments. Ptolemy allowed a ray of light to be incident upon a surface of water, like so:
air
water
air
water
This intrigued Ptolemy. He would set up a careful experiment to test a hypothesis he had made about an oar appearing bent in water, with consistent increments to see how data would fit when plotted together. In fact, he would derive a relationship that he thought seemed consistent with patterns he had seen in his astronomical work. But- and this is an important “but”- the reason his pattern emerged was because of selective rounding rules, where he would round some data one way, and other data another way. As a result, the relationship he derived was incorrect, though close to the actual relationship.
This was not altogether his fault, of course; he had done work in astronomy, and in this work, he had noticed a similar type of pattern, and so thought that nature had a unity in this sense. It did not, but confirmation bias would dominate Ptolemy’s thinking on this matter. What he did next, however, was to stop searching for the answer to why light behaves in this way, because he thought that he had found the answer.4 Nor, in fact, did any of his contemporaries. This shows how necessary all components of the scientific community are: yes, we need primary researchers that determine new relationships, but we also need reviewers and secondary researchers that try to check or replicate those relationships. Without both working together, we could find all the wrong relationships and think- because they give good results- that they are wholly correct.
Following this work, though, a line of scientists would study the relationships that Ptolemy had seen, each producing wholly correct results that are consistent with modern study of optics. For instance, Ibn al-Haitham (a.k.a. Alhazen) would study Ptolemy’s work and produce work in the early 1200s CE toward a refraction law. Qutb al-Din al-Shirazi and Kamal al-Din al-Farisi would be among the first to study and trace the path of light through large spheres of water, becoming some of the first to explain the optics of a rainbow- and not only this, but a double rainbow, and the first observation of a triple or tertiary rainbow in 13045. Theodoric of Freiburg, a French-German monk, would conduct work in this area, and his surviving diagrams are remarkable in their consistency with modern understanding of rainbows.
The first accurate model describing Ptolemy’s work on the bending of light, however, would come in 984 CE, and would deal with the small inconsistencies in Ptolemy’s data and model. Ibn Sahl of Baghdad, a Muslim scientist living in the time of the Islamic Golden Age, would derive this relationship:
𝑛𝑖sin 𝜃𝑖 = 𝑛𝑟sin 𝜃𝑟
4
Wilk, S.R. “Claudius Ptolemy’s Law of Refraction.” Optics and Photonics, 2004, p. 14-17
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Where 𝜃𝑖 is the angle of light incident upon a boundary (with respect to a normal line), and 𝜃𝑟 would be the angle of refraction of light moving into a new medium (with respect to the same normal line). The 𝑛’s each stand for the index of refraction, which will be explained
momentarily.
This is the correct formulation that Ptolemy could have derived, but the data fit so well with his quadratic relationship, and with discrepancies only becoming major at very large angles, it was hard to know that this was the true relationship.
The work of Ibn Sahl would not reach the European world. Instead, it would take until 1602 when Ibn Sahl’s work was rediscovered. Though unpublished, two prominent scientists of the time- Thomas Harriot and Johannes Kepler- would discuss it in letters. It would not appear in a European journal until 1621, when Dutch mathematician Willebrord van Roijen Snellius derived an equivalent formula, though it would not be published in his lifetime. René Descartes (yes, the same!) would derive the law in his own 1637 essay Dioptrics, and would also use this
relationship to understand rainbows. Pierre de Fermat would arrive at the same result soon after Descartes with his now-familiar principle of least time. Christiaan Huygens and Augustin-Jean Fresnel would use the wave nature of light to show why this relationship works, which we will discuss in a moment.
As for why it is called Snell’s Law? My best guess is that at the time of Descartes, several of his contemporaries- including Fermat and Huygens would accuse Descartes of plagiarizing Snell’s work as his own (though there has been little evidence to suggest this was the case). Because of this, the popularization of Snell’s name with the relationship came to cement the law as his, rather than Descartes’ Law, or Sahl’s Law, or even Ptolemy’s Law.
At any rate, it would be some time before it was understood what this index of refraction meant. It has been shown in more modern work that 𝑛 =𝑐
𝑣, where 𝑐 is the speed of light in a vacuum, and 𝑣 is the speed of light in a particular medium. Why does light slow down when traversing a medium? In a logical sense, there is more “stuff” in the way, when compared to a vacuum or even air. Because there is more matter, it is more likely for photons (pieces of light) to become absorbed by the material before being re-emitted and allowed to carry on, extending the time taken to travel through the material. (I am hand-waving on this explanation. For a more detailed mathematical explanation, consult your nearest optics textbook!)
To give you some idea of what values of 𝑛 you might deal with, typical indices of refraction for everyday materials range from 1.001 (like air) to about 2.4 (like diamonds) to as high as 3.45 (silicon) and 4.0 (germanium). The higher the index of refraction, the slower the speed of light in that particular medium. This, coupled with Fermat’s principle of least time, leads us to
𝜃𝑖
𝑛𝑖 < 𝑛𝑟
𝑛𝑟
𝜃𝑟
This is what the law of refraction predicts. Let us see why by stripping the diagram bare and considering two points, one on each side of the boundary:
𝐴 𝑛𝑖 < 𝑛𝑟
𝑛𝑟
𝐵
What path should light take between 𝐴 and 𝐵, such that we minimize the time taken to traverse this distance? Let us consider three possible paths:
𝐼𝐼 𝐼𝐼𝐼
𝐼
Path one moves toward the boundary almost immediately, and spends a lot of time in the new medium with a higher refraction index. Path two moves straight across, in a line, but does spend some time in the new medium. The third path, however, minimizes the time spent in the new medium. If the new medium has a higher refractive index, this means light moves slower in this medium. As a result, if we want to minimize the time taken to get from 𝐴 to 𝐵, Path III is preferred. (Again- I am hand-waving on this derivation. For more detailed mathematical analysis, consult your nearest optics textbook! Or wait until version 1.0 of these notes are collected together into one book, when I will be much more thorough than before.)
As we can see, the light bends away from the normal line, for precisely the same reasons as before: minimizing the time taken to travel between two points on either side of the boundary. The bending is described by the refraction law. But, there will come a point when the incidence angle will result in refracted light bending across the boundary, instead of actually transmitting through into the new medium:
This is a critical angle (𝜃𝑐), and is the angle at which total internal reflection happens- meaning all of the light meeting the boundary is instead reflected back into the same boundary.
So, what does that mean for our bent straw, like we talked about at the beginning of the section? We can answer that and more by considering a new situation: suppose I took you fishing by a lake, and asked you to spear a fish. Below is a diagram of where the fish appears, to your eye:
Where should you aim your spear? Directly at the fish, or somewhere different? Knowing what we know about refraction, let us consider this carefully. If we see the light from that fish as coming from that location, let us draw a light ray signifying this, from the boundary to our eye:
Here, I have filled in the actual location of the fish: slightly closer to, and slightly below, where we see the fish. This is where you should aim:
That is, if you want to eat that night. If you aim at the image of the fish (or where your eyes perceive the fish), you will miss your target. Guaranteed, every time.
Now that we have some understanding of refraction, we can begin studying what happens with more interesting shapes. We will turn again to our old, familiar physicist, Sir Isaac Newton.
Subsection: Mirages
Before that, however, we must look at an everyday aspect of refraction that we usually
experience in the summer. Have you ever looked down the road, and it seemed as though there was water scattered across the road… but then, when you move to that location, there is no water to be found? You are seeing a mirage. The reason this happens has to do with the refraction of light.
You should have learned in chemistry that one of the factors that can affect the density of an object (particularly a liquid or a gas) is its temperature. The higher the temperature of the fluid, the less dense it becomes. On a hot day, the air near to the ground has a much higher temperature than the air just above that, so the density of air lower to the ground is lower than that of air higher up. As a result, the refractive index changes! The refractive index for a less-dense fluid is less than a more-dense version of the same fluid. Light leaving objects will bend toward your eye, in addition to traveling straight across. This diagram shows what I mean:
You Object “arrow”
This means that, in essence, you are not seeing water when you see a mirage- you are seeing a mirror-like-“reflection” of the sky! (Next time you have a chance to see this, look at the shapes around the image- you should see inverted trees, road signs, cars, buildings, etc in the “mirage pool.”)
Newton’s Prisms, Dispersion, and Rainbows
Newton has been with us since the beginning of the course. From his laws of motion, to his law of gravitation that synthesized Kepler’s Laws, to his development of a whole field of
mathematics (calculus) to solve these problems, he truly is the Grandfather of physics, and- arguably- one of the greatest intellects ever to live. In his time putting together his Principia, he also had time to put together his treatise called Opticks, which laid down some of the
fundamental relationships for modern optics. Where Descartes, Snellius, Theodorick, Shirazi, and Ptolemy developed basic understanding of the refraction of light and many on this list were able to describe the optics of rainbows, Newton was one of the first to describe why rainbows formed, in addition to being the first to describe how lenses should be constructed. For all his great accomplishments, this was almost a “victory lap” of sorts.
Oh, and he also did not turn 26 years of age until after all of these achievements.
What he found is probably already known to you: he separated light into the spectrum. He was among the first to write down and formally understand that light exists as a spectrum, that white light has constituent parts that can be separated and added together. The reason for this is that the refraction of light depends not only on the material light is passing through; it also depends on the specific wavelength of light.
Newton took a glass triangular prism and shone white light through it. When this happened, this is what he saw:6
The figure shows sunlight proceeding into the prism from the right, and separating into the spectrum of colors: red, orange, yellow, green, blue, and violet. (Newton inserted indigo.7) This
6
effect is now called the dispersion of light, owing to the slightly different indices of refraction. For example: in crown glass, the index for red light is 1.50917, while the index for blue light is 52136.8 A small difference, but distinct enough to cause this separation.
Subsection: The Green Flash of Sunset
Because the refraction of light depends on the wavelength of light, there is a very neat optical effect. Every sunrise and sunset, at the very moments surrounding when the sun first breaches the horizon or finishes setting, you can observe a “green flash” at the horizon in the direction of the sun. As the sunlight refracts through the atmosphere and scatters away (see later sections), we see a red sunrise/sunset, as the other parts of the spectrum are “subtracted away.” But, at the instance of setting, some green light does make it through the atmosphere, and because red light has a different index of refraction in air than green light, you can actually see two images of the sun: a red one, and a green one, offset a little higher. This green sun will appear to set above the horizon, and is a faint, yet exhilarating, sight to see.
In order to see this phenomenon, though, you need a partner. You must stand with your eyes closed (I suggest holding hands with your partner), looking down at the ground, whatever you can do to get your eyes dark-adjusted. Your friend looks toward the horizon, and when the sun is about to set (or rise), they give you the cue to look, and you will be able to open your eyes and see it.
It is something that can only be experienced properly live. This summer, I charge you to see the flash with your own eyes!
Subsection: The Discovery of Infrared and Ultraviolet Light
Now, following Newton’s work “Opticks,” German astronomer Frederick William Herschel would experiment with prisms and try to measure the “temperature” of each color in the visible spectrum. What he found, however, was that there was a range where the temperature was warmer than room temperature, but it did not have a corresponding color. This was of a
wavelength below red, which he dubbed “calorific rays” in his paper in 1800. These would later be known as infrared rays.
More interesting still was that this part of the spectrum had been predicted to exist by Émilie du Châtelet in 1737, when she was studying fire. Herschel’s work would finally confirm her prediction.
Following the discovery of infrared light, German physicist Johann Wilhelm Ritter would ask himself this question: “if there was light beyond red, is there light beyond violet?” He would find that separating out such rays and shining them on a paper soaked in silver chloride would react
7
It would seem, for numerological reasons. It is possible that what he called blue we would now call cyan, which we do not now distinguish from green and blue as a distinct color. We do know that he associated his ROYGBIV spectrum with the seven notes on a musical scale, and saw some sort of divine pattern in this. However, in the present day, we tend not to list indigo as a distinct color on the spectrum.
8
much more quickly than violet light alone, and so called them “oxidizing rays” in his paper in 1801. The name would evolve over time, eventually settling on ultraviolet, and by a hundred years later, it would be known that ultraviolet light would be useful in sterilizing bacteria, leading to advances in medical treatments and patient care.
Color and the Appearance of Objects
Now, we ask a more basic question: Why is it that objects appear the way they do, at all? Why is the grass green, for example?
When the Sun emits light across the whole spectrum, a fair amount of that light is emitted in the visible portion of the electromagnetic spectrum, with colors from red to violet. We perceive that whole spectrum added together as white light. That light will interact with molecules in the grass, which will absorb different wavelengths of light and re-emit others. Whatever is absorbed is no longer part of the spectrum coming to you from that object. In the case of grass, chlorophyll absorbs both blue and red light, which means that of the “ROYGBV” spectrum, the colors that are left are mostly “YG,” hence why the grass is green. Any object you see around you is the color that it appears to be because only those wavelengths are reaching your eyes; the rest are being absorbed. (So, in a manner of speaking, grass is not green, but is actually “every color but green,” as green is (mostly) the only color not being absorbed by grass.
Subsection: Rayleigh Scattering, a.k.a. “Physics for Five-Year-Olds”
This, then, begs the question that has plagued us since we were five years old: “Why is the sky blue?”9
We might be tempted to say “it’s because the sky only reflects blue light,” but that is not actually the case. We might also be tempted to say that it is because the ocean is blue, but the ocean is made of water; last I checked, water by itself is fairly clear. The ocean is blue because the sky is blue, not the other way around.
So… why is the sky blue?
The story is fascinating10, but we will give the “Cliff notes” version here. In 1859, Irish physicist John Tyndall was trying to purify air in order to study infrared radiation. He determined that if you were to shine an intense light on a container of air, if it was contaminated with particles, light would scatter off of them. He also noticed, however, that when this would happen, blue light tended to scatter disproportionately. This would make the container would give off a blue glow, while the light that passed through the other side appeared reddish in color. This meant that the blue must have been subtracted from the spectrum of light as it was scattered away through the container! This would become known as the Tyndall Effect or Tyndall scattering.
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We all have probably annoyed our parents with this. If your experience was similar to mine, the response was likely, “just eat your French fries.”
10
Soon after in 1904, however, John William Strutt, a.k.a. the 3rd Baron Rayleigh (“Lord
Rayleigh”) in England, would find that atmospheric scattering was not quite the Tyndall effect, though they shared similar mathematical relationships. The Tyndall effect requires that the size of the particles be on the order of a single wavelength of light, while Rayleigh scattering- a better description of atmospheric scattering- required the particles involved to be somewhat smaller, on the order of 40 or more times smaller than the wavelength of light.
What is actually happening in either event is very intriguing: when light impacts upon an atom, it will excite an electron to a higher energy state, making it vibrate with the same frequency as the electric field carried by light. This vibrating electric field makes the atom emit the same color light as was absorbed, in a different and random direction.
Both effects say that the amount of scattering happens depends on the wavelength; specifically, the amount of scattering goes as 1
𝜆4, meaning the smaller the wavelength, the more scattering happens for that wavelength. Because of this, blue light scatters much more than red light, and does so for particles that are up to about a tenth of a micron (or 𝜇𝑚), or ~10,000 times smaller than the width of one of the hairs on your head.
The overall effect that describes Tyndall and Rayleigh is called Mie scattering, which is also known as “Lorenz-Mie-Debye solution” to Maxwell’s equations for electromagnetism. This was arrived at by Gustav Mie, another German physicist, and independently derived at the same time by Danish physicist Ludwig Lorenz and Dutch-American physicist/chemist Peter Debye, which is why they are all named in the solution.
Now, those of you who are alert might ask yourself a question: “Sure, I accept that if scattering goes as 𝜆−4, then the sky should be blue, and not red. But, then, why is it not violet, which has an even shorter wavelength?”
There is a combination of effects at work, here. One is that our eyes are sensitive to only three regimes of color: red, blue, and yellow, with a set of light-sensitive cells called “rods” and “cones” in each eye. Because we are more sensitive to blue than we are to violet, it is easier for us to see blue than it is for us to see red. The other is that if you look at the emission curve of the Sun- that is, how intensely light of different wavelengths is emitted from our star- less violet light is emitted from the Sun than blue light. These two effects together produce a blue sky to our eyes, rather than a violet sky.
On other planets orbiting other stars, with different atmospheric compositions, they may have different colored skies. For us, it is blue!
Now, why are sunsets and sunrises so brilliantly red? Think: if farther away from you, toward the horizon, all the blue light is getting scattered away, that means only the redder wavelengths are left for you to see!
So, to recap our five-year-old questions:
- Why is the sky blue? Rayleigh or Mie scattering, because the scattering goes as 𝜆−4. - Why isn’t the sky violet? Our eyes aren’t as sensitive to violet light, and the Sun doesn’t
produce as much violet light as it does blue light.
- Why are clouds white? Rayleigh or Mie scattering occurs when the particle sizes are very small. Water particles are larger, so light of different wavelengths scatter more equally. All the colors added together give you white.
- Why are sunsets/sunrises red? Because far away from you, but still on Earth, the blue light has been scattered away, leaving only the red light for you to see.
Lenses and the Lens Equation
Now, we ask an altogether different question: we have allowed light to pass through things before, in our refraction section. But what if we shape the media through which light passes to specific shapes? We can adjust very precisely how light focuses. This is especially helpful for photography and film projects, but also for the human eye.
As we will discuss later, the human eye is imperfect and not actually very good at doing what it is supposed to do; it is simply “good enough” to allow us to survive in our environment. But, often, even this passable imperfection is even more imperfect than usual, and we need to correct this with optical lenses, or glasses. There are two broad types of lenses we will discuss in this course, and they share names with the types of mirrors we have learned about: concave and convex. They take these shapes:
convex concave
What sets lenses apart from mirrors is, obviously, that light can pass through them. As a result, each lens will actually have two focus points.
We will also assume all lenses we deal with are symmetrical, though they do not have to be. It will make understanding how these lenses operate easier to understand, from a conceptual standpoint. You will likely not see asymmetric or “thick” lenses until your sophomore, junior, or senior year of college, if you pursue engineering, mathematics, or physics as your major.
With that aside, let us see how an image might form in each lens. First, we will study the convex lens.
Here, I have labeled both focal points (of length 1.70 𝑐𝑚) and radii of curvature, corresponding to each side of the lens, with an object positioned 3.71 𝑐𝑚 away from the lens. We begin drawing our ray diagram as usual, using the same three rays as before- only this time, the light will focus with respect to a focal point that is on the other side of this convex lens. (This will let us see why these are often called “converging” lenses.)
What is especially handy about lenses is that the mirror equation- through similar derivations as before- also applies, here! With lenses, however, we adopt the convention that object distances are always positive, and focal lengths/image distances on the opposite side of the lens are positive. Focal lengths and image distances on the same side of the lens as the object will be treated as negative. Doing the mathematics, we find that the image should form at a distance of
3.21 𝑐𝑚 away from the lens. When I measure the position on the diagram with Word, the distance comes to 3.18 𝑐𝑚, an error of about 0.9%.
Here, the light rays will diverge from the focus point in front of the lens. To see why, try drawing sections of the lens and treating them as a boundary between air and glass, then draw the
refraction diagrams. You should see the light moving away from that focal point. We continue:
This second ray must be drawn toward the focal point on the opposite side of the lens, as this is the focal point that will be diverging that light ray:
The third ray is drawn straight through the center of the lens, and the light rays extend to find the image:
Following the lens equation, the image distance should be −1.16 𝑐𝑚 away from the lens. Measuring in Word, I find the distance to be 1.22 𝑐𝑚. This corresponds to an error of 5% error. Again, not bad, considering the limitations of the software!
The general rules for concave and convex lenses are the same, just reversed, as those for concave and convex mirrors. Concave lenses and convex mirrors always form virtual images, while convex lenses and concave mirrors can form real images (and can project them to infinity, or focus light from infinitely far away on their focal points).
You have seen a convex lens before: in magnifying glasses! In addition, both types of lenses are used as corrective eyewear. To see why, consult the next section.
Now, we can think: how does our eye work? First, we need to understand what kind of lens our eyes use. We know that in order to form a real image, we need a convex lens. Our optical nerves need real images in order to have light stimulate them like a screen, so we therefore know that our optical lenses are convex. But this leads to an interesting conclusion: we see the world upside down. By that, I do not mean that the world is upside down, and that down is up, and so on- I mean that the images we perceive are upside down, and our brain automatically inverts them. Any time that our eyes are open, our brains are flipping the images we see so that we can make sense of the world with our eyes.
Part of the reason that babies have a hard time seeing far away and making sense of the world is that their eyes have not yet fully developed, so their brains do not yet understand the task that they will have to tirelessly perform throughout the human’s life.
Now, what if our lenses do not form perfectly, or are damaged later in life? We are probably familiar with the terms nearsightedness (myopia), farsightedness (hyperopia), and astigmatism. These all describe common conditions of the eye that can be corrected with proper prescription lenses. This is what is happening from an optical standpoint in each of the “sightedness” cases, though individual cases may deviate slightly from these diagrams:
In the case of myopia, the light is focusing at a point before the optical nerves, while in the case of hyperopia, the light is focusing at a point beyond the optical nerves. This can be corrected by the use of a diverging (concave) or converging (convex) lens, respectively. The reason this works is that, by making the light slightly diverge before hitting a myopic eye, or by making the light converge slightly before hitting a hyperopic eye, the light will focus at the proper place: the back of the eye.
Something altogether more interesting is happening in the case of an astigmatism. In this situation, the eye has become deformed in a kind of “football”-esque shape. Consider this diagram and description of astigmatism from 1911:11
11
What this diagram and captions are saying is that, in different planes, light passing through the eye is focusing at different locations, leading to a smeared out picture of the world. A person with astigmatism staring at one streetlight may actually see several lightbulbs at once, because of this effect- each with differing degrees of clarity. There is no simple way to describe a fix for this condition, other than to say that the lenses must be tailor-made for the person with the condition.
Pinhole Cameras
Multiple Lenses: Telescopes and Microscopes
